Q.1. Test the divisibility of following numbers by 2 :
(i) 2650
(ii) 69435
(iii) 59628
(iv) 789403
(v) 357986
(vi) 367314
Ans.
(i) The given number = 2650
Digit at unit’s place = 0
∴ It is divisible by 2.
(ii) The given number = 69435
Digit at unit’s place = 5
∴ It is not divisible by 2.
(iii) The given number = 59628
Digit at unit’s place = 8
∴ It is divisible by 2.
(iv) The given number = 789403
Digit at unit’s place = 3
∴ It is not divisible by 2.
(v) The given number = 357986
Digit at unit’s place = 6
∴ It is divisible by 2.
(vi) The given number = 367314
Digit at unit’s place = 4
∴ It is divisible by 2.
Q.2. Test the divisibility of following numbers by 3 :
(i) 733
(ii) 10038
(iii) 20701
(iv) 524781
(v) 79124
(vi) 872645
Ans.
(i) The given number = 733
Sum of its digits = 7 + 3 + 3 = 13, which is not divisible by 3.
∴ 733 is not divisible by 3.
(ii) The given number = 10038
Sum of its digits = 1 + 0 + 0 + 3 + 8
= 12, which is divisible by 3
∴ 10038 is divisible by 3.
(iii) The given number = 20701
Sum of its digits = 2 + 0 + 7 + 0 + 1
= 10, which is not divisible by 3
∴ 20701 is not divisible by 3.
(iv) The given number = 524781 Sum of its digits = 5 + 2 + 4 + 7 + 8 + 1 = 27, which is divisible by 3
∴ 524781 is divisible by 3.
(v) The given number = 79124 Sum of its digits = 7 + 9 + 1 + 2 + 4 = 23, which is not divisible by 3
∴ 79124 is not divisible by 3.
(vi) The given number = 872645 Sum of its digits = 8 + 7 + 2 + 6 + 4 + 5 = 32, which is not divisible by 3
∴ 872645 is not divisible by 3.
Q.3. Test the divisibility of the follo wing numbers by 4 :
(i) 618
(ii) 2314
(iii) 63712
(iv) 35056
(v) 946126
(vi) 810524
Ans. (i) The given number = 618
The number formed by ten’s and unit’s digits is 18, which is not divisible by 4.
∴ 618 is not divisible by 4.
(ii) The given number = 2314
The number formed by ten’s and unit’s
digits is 14, which is not divisible by 4.
∴ 2314 is not divisible by 4.
(iii) The given number = 63712
The number formed by ten’s and unit’s digits is 12, which is divisible by 4
∴ 63712 is divisible by 4.
(iv) The given number = 35056
The number formed by ten’s and unit’s digits is 56, which is divisible by 4.
∴ 35056 is divisible by 4.
(v) The given number = 946126
The number formed by ten’s and unit’s digits is 26, which is not divisible by 4.
∴ 946126 is not divisible by 4.
(vi) The given number = 810524
The number formed by ten’s and unit’s digits is 24, which is divisible by 4.
∴ 810524 is divisible by 4.
Q.5. Test the divisibility of the follo wing numbers by 6 :
(i) 2070
(ii) 46523
(iii) 71232
(iv) 934706
(v) 251730
(vi) 9087248
Ans.
(i) The given number = 2070
Its unit’s digit = 0
So, it is divisible by 2
Sum of its digits = 2 + 0 + 7 + 0 = 9,
which is divisible by 3
∴ The given number is divisible by 3.
So, 2070 is divisible by both 2 and 3.
Hence it is divisible by 6.
(ii) The given number = 46523
Its unit’s digit = 3
So, it is not divisible by 2
Hence 46523 is not divisible by 6.
(iii) The given number = 71232
Its unit’s digit = 2
So, it is divisible by 2
Sum of its digits = 7 + 1 + 2 + 3 + 2
= 15, which is divisible by 3
∴ 71232 is divisible by both 2 and 3
Hence it is divisible by 6.
(iv) The given number = 934706
Its unit’s digit = 6 So,
it is divisible by 2
Sum of its digits = 9 + 3 + 4 + 7 + 0 + 6
= 29, which is not divisible by 3
Hence 934706 is not divisible by 6.
(v) The given number = 251730
Its unit’s digit = 0
So, it is divisible by 2
Sum of its digits = 2 + 5 + 1 + 7 + 3 + 0
= 18, which is divisible by 3
∴ 251730 is divisible by both 2 and 3.
Hence it is divisible by 6.
(vi) 872536 is not divisible by 6 as sum of
its digits is 8 + 7 + 2 + 5 + 3 + 6 =31
which is not divisible by 3
Q.6 . Test the divisibility of the follo wing numbers by 7 :
(i) 826
(ii) 117
(iii) 2345
(iv) 6021
(v) 14126
(vi) 25368
Ans. We know that a number is divisible by the difference between twice the ones digit and the number formed by the other digits is either 0 or a multiple of 7
(i) 826, 6 × 2 = 12 and 82
Difference between 82 and 12 = 70
Which is divisible by 7
∴ 826 is divisible by 7
(ii) In 117, 7 × 2 = 14, 11
Difference between 14 and 11 = 14 – 11
= 3
Which is not divisible by 7
∴ 117 is not divisible by 7
(iii) In 2345, 5 × 2 = 10 and 234
Difference between 234 – 10 = 224
which is divisible by 7
∴ 2345 is divisible by 7
(iv) In 6021, 1 × 2 = 2, and 602
Difference between 602 and 2 = 600
which is not divisible by 7
∴ 6021 is not divisible by 7
(v) In 14126, 6 × 2 = 12 and 1412
Difference between 1412 – 12 = 1400
which is divisible by 7
∴ 14126 is divisible by 7
(vi) In 25368, 8 × 2 = 16 and 2536
Difference between 2536 and 16 = 2520
which is divisible 7
∴ 25368 is divisible by 7
Q.7. Test the divisibility of the follo wing numbers by 8 :
(i) 9364
(ii) 2138
(iii) 36792
(iv) 901674
(v) 136976
(vi) 1790184
Ans.
(i) The given number = 9364
The number formed by hundred’s, ten’s and unit’s digits is 364, which is not divisible by 8.
∴ 9364 is not divisible by 8.
(ii) The given number = 2138
The number formed by hundred’s, ten’s and unit’s digits is 138, which is not divisible by 8.
∴ 2138 is not divisible by 8.
(iii) The given number = 36792
The number formed by hundred’s, ten’s and unit’s digits is 792, which is divisible by 8.
∴ 36792 is divisible by 8.
(iv) The given number = 901674
The number formed by hundred’s, ten’s and unit’s digits is 674, which is not divisible by 8.
∴ 901674 is not divisible by 8.
( v ) The given number = 136976
The number formed by hundred’s, ten’s and unit’s digits is 976, which is divisible by 8.
∴ 136976 is divisible by 8.
(vi) The given number = 1790184
The number formed by hundred’s, ten’s and unit’s digits is 184, which is divisible by 8.
∴ 1790184 is divisible by 8.
Q.8. Test the divisibility of the following numbers by 9 :
(i) 2358
(ii) 3333
(iii) 98712
(iv) 257106
(v) 647514
(vi) 326999
Ans.
We know that a number is divisible by 9, if the sum of its digits is divisible by 7
(i) In 2358
Sum or digits : 2 + 3 + 5 + 8 = 18 which is divisible by 9
∴ 2358 is divisible by 9
(ii) In 3333
Sum of digit 3 + 3 + 3 + 3 = 12 which is not divisible by 9
∴ 3333
(iii) In 98712
Sum of digits = 9 + 8 + 7 + 1 + 2 = 27
Which is divisible by 9
∴ 98712 is divisible by 9
(iv) In 257106
Sum of digits = 2 + 5 + 7 + 1 + 0 + 6 = 21 which is not divisible by 9
∴ 257106 is not divisible by 9
(v) In 647514
Sum of digits = 6 + 4 + 7 + 5 + 1 + 4 = 27 which is divisible by 9
∴ 647514 is divisible by 9
(vi) In 326999
Sum of digits = 3 + 2 + 6 + 9 + 9 + 9 = 38 which is not divisible by 9
∴ 326999 is divisible by 9
Q.9. Test the divisibility of the following number by 10 :
(i) 5790
(ii) 63215
(iii) 55555
Ans. We know that a number is divisible by 10 if its one digit is 0
∴ (i) 5790 is divisible by 10
Q.10. Test the divisibility of the following numbers by 11 :
(i) 4334
(ii) 83721
(iii) 66311
(iv) 137269
(v) 901351
(vi) 8790322
Ans.
(i) The given number = 4334
Sum of its digits in odd places = 4 + 3 = 7
Sum of its digits in even places = 3 + 4 = 7
Difference of the two sums = 7 – 7 = 0
∴ 4334 is divisible by 11.
(ii) The given number = 83721
Sum of its digits in odd places = 1 + 7 + 8 = 16
Sum of its digits in even places = 2 + 3 = 5
Difference of the two sums = 16 – 5 = 11, which is multiple of 11.
∴ 83721 is divisible by 11.
(iii) The given number = 66311
Sum of its digits in odd places
= 1 + 3 + 6 = 10
Sum of its digits in even places
= 1 + 6 = 7
Difference of the two sums = 10 – 7 = 3, which is not a multiple of 11.
∴ 66311 is not divisible by 11.
(iv) The given number = 137269
Sum of its digits in odd places
= 9 + 2 + 3 = 14
Sum of its digits in even places
= 6 + 7 + 1 = 14
Difference of the two sums
= 14 – 14 = 0
∴ 137269 is divisible by 11.
(v) The given number = 901351
Sum of its digits in odd places
= 1 + 3 + 0 = 4
Sum of its digits in even places
= 5 + 1 + 9 = 15
Difference of the two sums = 15 – 4
= 11, which is a multiple of 11.
∴ 901351 is divisible by 11.
(vi) The given number = 8790322
Sum of its digits in odd places
= 2 + 3 + 9 + 8 = 22
Sum of its digits in even places
= 2 + 0 + 7 = 9
Difference of the two sums
= 22 – 9 = 13,
which is not a multiple of 11.
∴ 8790322 is not divisible by 11.
Q.11. In each of the following numbers , replace * by the smallest number to make it divisible by 3.
(i) 27*4
(ii) 53*46
(iii) 8*711
(iv) 62*35
(v) 234*17
(vi) 6*1054
Ans. (i) The given number = 27*4
Sum of its digits = 2 + 7 + 4 = 13
The number next to 13 which is divisible by 3 is 15.
∴ Required smallest number = 15 – 13
= 2.
(ii) The given number = 53*46
Sum of the given digits = 5 + 3 + 4 + 6
= 18, which is divisible by 3.
∴ Required smallest number = 0.
(iii) The given number = 8*711
Sum of the given digits = 8 + 7 + 1 + 1 = 17
The number next to 17, which is divisible by 3 is 18.
∴ Required smallest number = 18 – 17
= 1.
(iv) The given number = 62*35
Sum of the given digits = 6 + 2 + 3 + 5
= 16
The number next to 16, which is divisible by 3 is 18.
∴ Required smallest number = 18 – 16
= 2
(v) The given number = 234*17
Sum of the given digits
= 2 + 3 + 4 + 1 + 7 = 17
The number next to 17, which is divisible by 3 is 18.
∴ Required smallest number
= 18 – 17 = 1.
(vi) The given number = 6* 1054
Sum of the given digits = 6 + 1 + 0 + 5 + 4 = 16
The number next to 16, which is divisible by 3 is 18.
∴ Required smallest number
= 18 – 16 = 2.
Q.12. In each of the following numbers, replace * by the smallest number to make it divisible by 9.
(i) 65*5
(ii) 2*135
(iii) 6702*
(iv) 91*67
(v) 6678*1
(vi) 835*86
Ans.
(i) The given number = 65*5
Sum of its given digits = 6 + 5 + 5 = 16
The number next to 16, which is divisible by 9 is 18.
∴ Required smallest number = 18 – 16
= 2.
(ii) The given number = 2*135
Sum of its given digits = 2 + 1 + 3 + 5
= 11
The number next to 11, which is divisible by 9 is 18.
∴ Required smallest number
= 18 – 11 = 7.
(iii) The given number = 6702*
Sum of its given digits
= 6 + 7 + 0 + 2 = 15
The number next to 15, which is divisible by 9 is 18.
∴ Required smallest number = 18 – 15 = 3
(iv) The given number = 91*67
Sum of its given digits = 9 + 1 + 6 + 7 = 23
The number next to 23, which is divisible by 9 is 27.
∴ Required smallest number = 27 – 23 = 4.
(v) The given number = 6678*1
Sum of its given digits
= 6 + 6 + 7 + 8 + 1 = 28
The number next to 28, which is divisible by 9 is 36.
∴ Required smallest number
= 36 – 28 = 8.
(vi) The given number = 835*86
Sum of its given digits
= 8 + 3 + 5 + 8 + 6
= 30
The number next to 30, which is divisible
by 9 is 36.
∴ Required smallest number
= 36 – 30 = 6.
Q.13. In each of the following numbers, replace * by the smallest number to make it divisible by 11.
(i) 26*5
(ii) 39*43
(iii) 86*72
(iv) 467*91
(v) 1723*4
(vi) 9*8071
Ans. (i) The given number = 26*5
Sum of its digits is odd places
= 5 + 6 = 11
Sum of its digits in even places = * + 2
Difference of the two sums
= 11 – (* + 2)
The given number will be divisible by 11 if the difference of the two sums = 0.
∴ 11 – (* + 2) = 0
11 = * + 2
11 – 2 = *
9 = *
∴ Required smallest number = 9.
(ii) The given number = 39*43
Sum of its digits in odd places
= 3 + * + 3 = * + 6
Sum of its digits in even places
= 4 + 9 = 13
Difference of the two sums
= * + 6 – 13 = * – 7
The given number will be divisible by 11,
if the difference of the two sums = 0.
∴ * – 7 = 0
* = 7
∴ Required smallest number = 7.
(iii) The given number = 86*72
Sum of its digits in odd places
= 2 + * + 8 = * + 10
Sum of its digits in even places
= 7 + 6 = 13
Difference of the two sums
= * + 10 – 13 = * – 3
The given number will be divisible by
11, if the difference of the two sums = 0.
∴ * – 3 = 0
* = 3
∴ Required smallest number = 3.
(iv) The given number = 467*91
Sum of its digits in odd places
= 1 + * + 6 = * + 7
Sum of its digits in even places
= 9 + 7 + 4 = 20
Difference of the two sums
= 20 – (* + 7)
= 20 – * – 7 = 13 – *
Clearly the difference of the two sums
will be multiple of 11 if 13 – * = 11
∴ 13 – 11 = *
2 = *
* = 2
∴ Required smallest number = 2.
(v) The given number = 1723*4
Sum of its digits in odd places
= 4 + 3 + 7 = 14
Sum of its digits in even places
= * + 2 + 1 = * + 3
Difference of the two sums
= * + 3 – 14 = * – 11
∴ The given number will be divisible by
11, if * – 11 is a multiple of 11, which is
possible if * = 0.
∴ Required smallest number = 0.
(vi) The given number = 9*8071
Sum of its digits in odd places
= 1 + 0 + * = 1 + *
Sum of its digits in even places
= 7 + 8 + 9 = 24
Difference of the two sums
= 24 – 1 – * = 23 – *
∴ The given number will be divisible by
11, if 23 – * is a multiple of 11, which is possible if * = 1.
∴ Required smallest number = 1.
Q.14. Test the divisibility of
(i) 10000001 by 11
(ii) 19083625 by 11
(iii) 2134563 by 9
(iv) 10001001 by 3
(v) 10203574 by 4
(vi) 12030624 by 8
Ans.
(i) The given number = 10000001
Sum of its digits in odd places
= 1 + 0 + 0 + 0 = 1
Sum of its digits in even places
= 0 + 0 + 0 + 1 = 1
Difference of the two sums = 1 – 1 = 0
∴ The number 10000001 is divisible by 11.
(ii) The given number = 19083625
Sum of its digits in odd places
= 5 + 6 + 8 + 9 = 28
Sum of its digits in even places
= 2 + 3 + 0 + 1 = 6
Difference of the two sums = 28 – 6
= 22, which is divisible by 11.
∴ The number 19083625 is divisible by 11.
(iii) The given number = 2134563
Sum of its digits = 2 + 1 + 3 + 4 + 5 + 6 + 3
= 24, which is not divisible by 9.
∴ The number 2134563 is not divisible by 9.
(iv) The given number = 10001001
Sum of its digits = 1 + 0 + 0 + 0 + 1 + 0 + 0 + 1 = 3, which is divisible by 3.
∴ The number 10001001 is divisible by 3.
(v) The given number = 10203574
The number formed by its ten’s and unit’s digits is 74, which is not divisible by 4.
∴ The number 10203574 is not divisible by 4.
(vi) The given number = 12030624
The number formed by its hundred’s, ten’s and unit’s digits = 624, which is divisible by 8.
∴ The number 12030624 is divisible by 8.
Q.15. Which of the following are prime numbers ?
(i) 103
(ii)137
(iii) 161
(iv) 179
(v) 217
(vi) 277
(vii) 331
(viii) 397
Ans.
103, 137, 179, 277, 331, 397 are prime numbers.
Q.16. Give an example of a number
(i) Which is divisible by 2 but not by 4.
(ii) Which is divisible by 4 but not by 8.
(iii) Which is divisible by both 2 and 8 but not divisible by 16.
(v) Which is divisible by both 3 and 6 but not by 18.
Ans.
(i) 154
(ii) 612
(iii) 5112, 3816
(iv) 3426, 5142 etc.
Q.17. Write (T) for true and (F) for false against each of the followings statements :
(i) If a number is divisible by 4, it must be divisible by 8.
(ii) If a number is divisible by 8, it must be divisible by 4.
(iii) If a number divides the sum of two numbers exactly, it must exactly divide
the numbers separately.
(iv) If a number is divisible by both 9 and 10, it must be divisible by 90.
(v) A number is divisible by 18, if it is divisible by both 3 and 6.
(vi) If a number is divisible by 3 and 7, it must be divisible by 21.
(vii) The sum of two consecutive odd numbers is always divisible by 4.
(viii) If a number divides three numbers exactly, it must divide their sum exactly.
Ans.
(i) False
(ii) True
(iii) False
(iv) True
(v) False
(vi) True
(vii) True
(viii) True.
PRIME FACTORIZATION
Prime Factor. A factor of a given number is called a prime factor if this factor is a prime number.
Prime Factorization. To express a given number as a product of prime factors is called a prime factorization of the given number.
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2. How can I find the factors of a given number? 
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